WASHINGTON—Almost 6.4 million people selected a health-care plan on the federal marketplace or were automatically re-enrolled in the first month of the Affordable Care Act’s open-enrollment season this fall, Health and Human Services Secretary Sylvia Mathews Burwell said Tuesday.

That figure compares with about 5.4 million people who had selected plans using the federal exchange in the original enrollment period of Oct. 1, 2013, through March 31, 2014.

High consumer demand in mid-December helped boost the total, which includes about 1.9 million new consumers who obtained private coverage through the federal exchange, Ms. Burwell told reporters.

She said 4.5 million people who got coverage through the federal site in 2014 were re-enrolled. Of those, more than 30% returned to the site to re-enroll while the rest were re-enrolled automatically, but she didn’t supply actual numbers.

The figures don’t include consumers who selected plans through state-based insurance marketplaces.

The sign-ups in the period of Nov. 15 through Dec. 19 indicate that the second enrollment under the ACA, which runs through Feb. 15, is off to “an encouraging start,” Ms. Burwell said.

“We still have a ways to go but we are headed in the right direction, the marketplace is stable, people have chosen to shop,” she said.

The sign-ups far outstrip the first month of enrollment last year, when HealthCare.gov was hobbled by technology problems. About 106,185 people selected a plan on the federal and state exchanges between Oct. 1, 2013, and Nov. 2, 2013. Of them, 79,391 selected plans through state exchanges and 26,794 in the federal marketplace.

In total, as of mid-October 2014—before the new enrollment period began—some 6.7 million people had paid-up health coverage through the health law’s state and federal exchanges.

HealthCare.gov, the federal exchange that serves 37 states, has been under scrutiny as many Republicans have sought to repeal or curtail the ACA. This fall’s enrollment has been seen as a critical test of the health law and federal website, which underwent upgrades and obtained new management after last year’s chaotic debut.

In one snafu, in the days leading up to the Dec. 15 deadline for changing or picking coverage that kicks in Jan. 1, about 500,000 consumers who contacted a call center didn’t get to speak to a representative but were asked to leave contact information. HHS officials said those people would still be allowed to have coverage that starts Jan. 1. The vast majority were reached by Dec. 18 by HHS workers, the agency said.

Millions of Americans who didn’t go back to the site to select new coverage have been automatically re-enrolled—and may be surprised by higher premiums next year because other plans may be less expensive or their premiums may have risen. Those whose plans are being discontinued will be moved into similar plans.

Some state-run insurance sites extended enrollment deadlines and in-person sign-up hours due to heavy last-minute demand. For HealthCare.gov users, the average call-center wait time in from Dec. 13 through Dec. 19 was about 23 minutes. That is higher than the overall nine-minute wait time from Nov. 15 through Dec. 19, according to HHS.

Aside from the current open-enrollment season, Ms. Burwell said an important U.S. Supreme Court case challenging the subsidies provided on the federal exchange hasn’t changed the department’s focus in encouraging consumers to sign up for coverage. The Supreme Court said it would hear oral arguments on March 4 in the lawsuit contesting tax credits to consumers who purchased health insurance through the federal exchange. An estimated 4.7 million Americans have received such subsidies.

“Nothing has changed in our open enrollment in terms of subsidies and assistance people can get,” said Ms. Burwell. “One can create any kind of hypothetical about anything,” she said of the risk that consumers could lose subsidies based on the court’s ultimate ruling.

Enrollment is expected to continue climbing closer to the Feb. 15 deadline. The Obama administration has said it expects up to 9.9 million people to have private coverage through the insurance exchanges in 2015, although that still falls short of the 13 million initially expected to enroll under an April 2014 projection by the Congressional Budget Office.

About 14 million people used the HealthCare.gov website between Nov. 15 and Dec. 19, according to HHS.